Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-8x-3y &= 8 \\ -x-y &= -4\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}8x+3y &= -8\\ -3x-3y &= -12\end{align*}$ Add the top and bottom equations. $5x = -20$ Divide both sides by $5$ and reduce as necessary. $x = -4$ Substitute $-4$ for $x$ in the top equation. $-8( -4)-3y = 8$ $32-3y = 8$ $-3y = -24$ $y = 8$ The solution is $\enspace x = -4, \enspace y = 8$.